When Separate Longitudinal and Event Analyses Give Conflicting Results

Why Separate Analyses Can Disagree

A longitudinal analysis asks how a biomarker changes over time. An event analysis asks how event occurrence relates to the biomarker. When both are conducted separately, each may use a different subset of participants and a different time reference. If participants with worsening health are more likely to miss later biomarker visits, the observed measurements need not represent the original cohort. The biomarker average can then move in a direction that does not describe typical within-person change. The biomarker–event association can also differ depending on whether it is calculated at baseline or only among people with a later measurement.

The central issue is selection into the observed sample. A measured value is available only for people who remain under biomarker observation at that time. If continued observation is related to health, the measured participants may differ systematically from those without a measurement. This is not fixed merely by analyzing the longitudinal data and event data in separate models: the two analyses can still represent different selected populations and answer different questions.

An event can end biomarker measurement because the participant dies or experiences another terminal event. Health-related dropout is different: the participant stops providing biomarker measurements, but event status may still be ascertained afterward. Keeping these situations distinct matters. The example below uses health-related dropout before a landmark visit, while event follow-up continues. The landmark biomarker values for those who dropped out are therefore unobserved, not values recorded after a terminal event.

A Hypothetical Cohort

Consider eight hypothetical participants followed from baseline to a scheduled six-month visit, then for events after that visit. A higher biomarker value indicates worse health. Four participants begin at 16 and four begin at 8. By six months, the marker would be 18 in the first group and 10 in the second if measured. However, three of the four participants in the first group stop biomarker follow-up before the visit because their health is worsening. All four participants in the second group and one participant in the first group have a measured six-month value. Event follow-up continues for everyone.

Why Selection Changes a Mean

The difference between an observed-sample mean and a cohort mean can be expressed directly. Let \(Y\) be a biomarker value at a fixed time, and let \(R\) indicate whether it is observed. Since \(E(R)=P(R=1)\) and \(E(YR)=P(R=1)E(Y\mid R=1)\),

Therefore,

This identity applies when \(P(R=1)>0\). It shows that the observed mean equals the cohort mean when observation is unrelated to the value in the covariance sense. If higher values are less likely to be observed, then \(\operatorname{Cov}(Y,R)<0\), and the observed mean is below the cohort mean. In the example, participants with a higher six-month marker are less likely to have a measurement, so the observed mean, 11.6, is below the all-participant mean of 14.

This is a statement about representation, not a claim that missing values can always be recovered. If the missing values are unknown, the full cohort mean is not identified from the observed mean alone. The value 14 in the example follows from stipulated hypothetical values; an analyst who only saw the five observed values would need additional assumptions or information to estimate the mean for all eight.

The Biomarker–Event Association Can Also Change

The same hypothetical cohort illustrates why separate event analyses can appear to conflict. At baseline, the higher-marker group has three events among four participants, while the lower-marker group has one among four. This is a positive baseline association between the marker group and subsequent event risk. But an analysis at the six-month landmark can include only participants who have a measured six-month marker. The three high-marker participants who dropped out are then excluded from the biomarker–event comparison, even though their event follow-up continues.

Baseline and landmark analyses answer different questions. The baseline comparison uses the original groups and counts events after baseline. The landmark comparison relates a later measured marker to events after the landmark among a selected subset with measurements. A landmark analysis may be useful when the scientific question is explicitly about prognosis among people reaching that time point with a measurement. It does not automatically estimate the original cohort association. Likewise, an analysis using only participants with complete longitudinal data estimates a selected-sample pattern unless the selection process can be justified as irrelevant to the target.

The earlier material on complete-case comparisons, survivor averages, and health-related attrition motivates these distinctions. Here the practical lesson is to report the sampling set and time origin alongside each result. A numerical biomarker–event association without that information is incomplete.

Assumptions, Diagnostics, and Limits

A complete-case longitudinal summary describes people who have measurements at the required times. To interpret it as representing a broader cohort trajectory requires a defensible reason that selection into complete follow-up does not distort the target summary, or a valid adjustment based on adequate observed information. Merely observing that the available measurements rise or fall does not establish this condition. Health-related dropout makes it particularly doubtful when health history is associated with both the biomarker and continued observation.

Useful diagnostics begin with the observed data rather than a single model coefficient. Tabulate the number measured at each visit; describe how dropout varies by baseline marker, earlier marker values, and recorded health indicators; and compare the baseline characteristics of participants with and without later measurements. For the event analysis, state how many people are included, the time origin, and whether participants who stopped biomarker follow-up remain included in event follow-up. These checks can reveal selection patterns, but they cannot prove that unobserved marker values follow any particular trajectory.

Several alternatives answer distinct questions. A paired analysis uses only participants measured at both times and describes their average within-person change. An available-case mean uses everyone measured at each visit but may compare different participant mixtures over time. A baseline event analysis preserves the original cohort grouping, while a landmark analysis asks about later event risk conditional on reaching the landmark and having the required data. None of these choices is universally correct: the appropriate analysis depends on the target question and the assumptions that can be supported. A joint modeling approach is motivated in part by the need to represent longitudinal and event processes together, but the examples here do not establish a particular joint model or guarantee that one resolves all missing-data problems.

A common mistake is to say, “The average marker fell, so participants’ health improved.” That conclusion confuses a change in the observed-sample mean with within-person change. In the worked cohort, the observed mean falls by 0.4 units, while every paired value rises by two units and the stipulated full-cohort mean rises by two. The correction is to report which participants contribute to each mean, examine how observation relates to health, and avoid assigning the observed-sample trend to the whole cohort without justification.